I agree with others that advised to stand your ground, stating that your' property and buildings on it are historical and that a $500 cash offer or tear down the shed for free, is a slap in the face. Advise the Tree Removing Company that they are to cease and desist all further tree removal and that you are obtaining a layer or going to see yours, if you already have a good one. They will likely have their Insurance Company take care of it and with the settlement, you can have the shed torn down and rebuilt using some of the salvaged wood and other items from it and the same wood from the area. You would then have a new but still historical building on the property as some of the old wood, hinges, nails, windows, latches and door knobs were used in the rebuild.
That is a beautiful home and barn you have!
Good luck!
Frank